Stabilizer Codes Research Articles

Memory-corrected quantum repeaters with adaptive syndrome identification

We address the challenge of incorporating encoded quantum memories into an exact secret key rate analysis for small and intermediate-scale quantum repeaters. To this end, we introduce the check matrix model and quantify the resilience of stabilizer codes of up to eleven qubits against Pauli noise, obtaining analytical expressions for effective logical error probabilities. Generally, we find that the five-qubit and Steane codes either outperform more complex, larger codes in the experimentally relevant parameter regimes or have a lower resource overhead. Subsequently, we apply our results to calculate lower bounds on the asymptotic secret key rate in memory-corrected quantum repeaters when using the five-qubit or Steane codes on the memory qubits. The five-qubit code drastically increases the effective memory coherence time, reducing a phase flip probability of 1% to 0.001% when employing an error syndrome identification adapted to the quantum noise channel. Furthermore, it mitigates the impact of faulty Bell state measurements and imperfect state preparation, lowering the minimally required depolarization parameter for nonzero secret key rates in an eight-segment repeater from 98.4% to 96.4%. As a result, the memory-corrected quantum repeater can often generate secret keys in experimental parameter regimes where the unencoded repeater fails to produce a secret key. In an eight-segment repeater, one can even achieve nonvanishing secret key rates up to distances of 2000 km for memory coherence times of tc=10 s or less using multiplexing. Assuming a zero-distance link-coupling efficiency p0=0.7, a depolarization parameter μ=0.99, tc=10 s, and an 800 km total repeater length, we obtain a secret key rate of 4.85 Hz, beating both the unencoded repeater that provides 1.25 Hz and ideal twin-field quantum key distribution with 0.71 Hz at GHz clock rates. Published by the American Physical Society 2025

Fast algorithms for classical specifications of stabiliser states and Clifford gates

The stabiliser formalism plays a central role in quantum computing, error correction, and fault tolerance. Conversions between and verifications of different specifications of stabiliser states and Clifford gates are important components of many classical algorithms in quantum information, e.g. for gate synthesis, circuit optimisation, and simulating quantum circuits. These core functions are also used in the numerical experiments critical to formulating and testing mathematical conjectures on the stabiliser formalism.We develop novel mathematical insights concerning stabiliser states and Clifford gates that significantly clarify their descriptions. We then utilise these to provide ten new fast algorithms which offer asymptotic advantages over any existing implementations. We show how to rapidly verify that a vector is a stabiliser state, and interconvert between its specification as amplitudes, a quadratic form, and a check matrix. These methods are leveraged to rapidly check if a given unitary matrix is a Clifford gate and to interconvert between the matrix of a Clifford gate and its compact specification as a stabiliser tableau.For example, we extract the stabiliser tableau of a 2n×2n matrix, promised to be a Clifford gate, in O(n2n) time. Remarkably, it is not necessary to read all the elements of a Clifford gate matrix to extract its stabiliser tableau. This is an asymptotic speedup over the best-known method that is exponential in the number of qubits.We provide implementations of our algorithms in Python and C++ that exhibit vastly improved practical performance over existing algorithms in the cases where they exist.

Eigenvalue Sensitivity Computations for Linear Stability Theory

To realize the drag reduction benefit of boundary-layer transition control strategies, it is crucial to integrate transition prediction into the vehicle design through an optimization process. The integration of transition prediction based on linear stability analysis into adjoint-based design optimization requires coupling an adjoint-enabled computational fluid dynamics (CFD) solver with an adjoint-enabled linear stability code. In particular, the boundary-layer transition location is often predicted using the N-factor method based on linear stability theory (LST). Thus, the sensitivity of the linear-stability eigenvalues constitutes an essential building block for optimizing the laminar flow performance. The present paper describes an implementation of LST eigenvalue sensitivity analysis that can be easily coupled with a CFD solver. Specifically, we describe a discrete adjoint formulation for the transition location prediction based on the N-factor method. The verification of this formulation is carried out by comparing the adjoint-based sensitivity of the local growth rate of a given instability mode with respect to the disturbance frequency and the adjoint-based sensitivity of the transition location with respect to the spanwise wavenumber with those sensitivities computed using a finite-difference approximation. Finally, the adjoint LST formulation is applied to flat-plate boundary-layer flows at transonic, supersonic, and hypersonic conditions to determine the behavior and sensitivities of the transition location with respect to a range of disturbance spanwise wavenumbers.

Nonunitary Gates Using Measurements Only.

Measurement-based quantum computation (MBQC) is a universal platform to realize unitary gates, only using measurements that act on a preprepared entangled resource state. By deforming the measurement bases, as well as the geometry of the resource state, we show that MBQC circuits always transmit and act on the input state but generally realize nonunitary logical gates. In contrast to the stabilizer formalism that is often used for unitary gates, we find that ZX-calculus is an ideal computation method for these nonunitary gates. As opposed to unitary gates, nonunitary gates cannot be applied with certainty, due to the randomness of quantum measurements. We maximize the success probability of realizing nonunitary gates and discuss applications including imaginary time evolution, which we demonstrate on a noisy intermediate-scale quantum device.

Characterization of stabilizier states and magic states in terms of Tsallis and Rényi entropies for qubit systems

Abstract Uncertainty relations are fundamental in quantum mechanics, distinguishing it from classical physics by setting limits on the precision of incompatible measurements. For qubit systems, considering the three incompatible Pauli observables, we propose two entropic quantifiers of uncertainty exploiting the Tsallis entropies and Rényi entropies. We analyze the minimum and maximum uncertainty states and uncover their close connections with some important families of states in the stabilizer formalism. Explicitly, our findings show that for the two entropic quantifiers of uncertainties with suitable parameters, the stabilizer states correspond to the minimum uncertainty states, while the T-type magic states correspond to the maximum uncertainty states. Additionally, we identify that the H-type magic states serve as the saddle points in the Tsallis entropy framework. These results expand the recent proposed characterizations of stabilizer and magic states via the Shannon entropic uncertainty relation [Li et al 2024 Phys. Scr. 99, 035117].

Flag Gadgets Based on Classical Codes

Fault-tolerant syndrome extraction is a key ingredient in implementing fault-tolerant quantum computation. While conventional methods use a number of extra qubits that are linear in the weight of the syndrome, several improvements have been introduced using flag gadgets. In this work, we develop a framework to design flag gadgets using classical codes. Using this framework, we show how to perform fault-tolerant syndrome extraction for any stabilizer code with arbitrary distance using exponentially fewer qubits than conventional methods when qubit measurement and reset are relatively slow compared to a round of error correction. In particular, our method requires only (2t+1)t⌈log2(w)⌉ flag qubits to fault-tolerantly measure a weight-w stabilizer. We further take advantage of the saving provided by our construction to fault-tolerantly measure multiple stabilizers using a single gadget and show that it maintains the same exponential advantage when it is used to fault-tolerantly extract the syndromes of quantum low-density parity-check codes. Using the developed framework, we perform computer-assisted search to find several small examples where our constructions reduce the number of qubits required. These small examples may be relevant to near-term experiments on small-scale quantum computers. Published by the American Physical Society 2024

Efficient local operations and classical communication extraction of quantum information encoded in stabilizer codes

Efficient local operations and classical communication extraction of quantum information encoded in stabilizer codes

Stabilizer code model with noninvertible symmetries: Strange fractons, confinement, and noncommutative and non-Abelian fusion rules

We introduce a stabilizer code model with a qutrit at every edge on a square lattice and with noninvertible plaquette operators. The degeneracy of the ground state is topological as in the toric code, and it also has the usual deconfined excitations consisting of pairs of electric and magnetic charges. However, there are novel types of confined fractonic excitations composed of a cluster of adjacent faces with vanishing flux. They manifest confinement, and even larger configurations of these fractons are fully immobile although they acquire emergent internal degrees of freedom. Deconfined excitations change their nature in presence of these fractonic defects. As for instance, fractonic defects can absorb magnetic charges making magnetic monopoles exist while electric charges acquire restricted mobility. Furthermore, some generalized symmetries can annihilate any ground state and also the full sector of fully mobile excitations. All these properties can be captured via a novel type of and fusion category in which the product is associative but does not commute, and can be expressed as a sum of (operator) equivalence classes. Generalized noninvertible symmetries give rise to the feature that the fusion products form a nonunital category without a proper identity. We show that a variant of this model features a deconfined fracton liquid phase and a phase where the dual (magnetic) strings have condensed. Published by the American Physical Society 2024

LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism

We present a novel formalism to both understand and construct mixers that preserve a given subspace. The method connects and utilizes the stabilizer formalism that is used in error correcting codes. This can be useful in the setting when the quantum approximate optimization algorithm (QAOA), a popular meta-heuristic for solving combinatorial optimization problems, is applied in the setting where the constraints of the problem lead to a feasible subspace that is large but easy to specify. The proposed method gives a systematic way to construct mixers that are resource efficient in the number of controlled not gates and can be understood as a generalization of the well-known X and XY mixers and a relaxation of the Grover mixer: Given a basis of any subspace, a resource efficient mixer can be constructed that preserves the subspace. The numerical examples provided show a dramatic reduction of CX gates when compared to previous results. We call our approach logical X-Mixer or logical X QAOA (LX-QAOA), since it can be understood as dividing the subspace into code spaces of stabilizers S and consecutively applying logical rotational X gates associated with these code spaces. Overall, we hope that this new perspective can lead to further insight into the development of quantum algorithms.

Quantum subsystem codes, CFTs and their ℤ2-gaugings

We construct Narain conformal field theories (CFTs) from quantum subsystem codes, a more comprehensive class of quantum error-correcting codes than quantum stabilizer codes, for qudit systems of prime dimensions. The resulting code CFTs exhibit a global ℤ2 symmetry, enabling us to perform the ℤ2-gauging to derive their orbifolded and fermionized theories when the symmetry is non-anomalous. We classify a subset of these subsystem code CFTs using weighted oriented graphs and enumerate those with small central charges. Consequently, we identify several bosonic code CFTs self-dual under the ℤ2-orbifold, new supersymmetric code CFTs, and a few fermionic code CFTs with spontaneously broken supersymmetry.

Group frames via magic states with applications to SIC-POVMs and MUBs

We connect magic (non-stabilizer) states, symmetric informationally complete positive operator valued measures (SIC-POVMs), and mutually unbiased bases (MUBs) in the context of group frames, and study their interplay. Magic states are quantum resources in the stabilizer formalism of quantum computation. SIC-POVMs and MUBs are fundamental structures in quantum information theory with many applications in quantum foundations, quantum state tomography, and quantum cryptography, etc. In this work, we study group frames constructed from some prominent magic states, and further investigate their applications. Our method exploits the orbit of discrete Heisenberg–Weyl group acting on an initial fiducial state. We quantify the distance of the group frames from SIC-POVMs and MUBs, respectively. As a simple corollary, we reproduce a complete family of MUBs of any prime dimensional system by introducing the concept of MUB fiducial states, analogous to the well-known SIC-POVM fiducial states. We present an intuitive and direct construction of MUB fiducial states via quantum T-gates, and demonstrate that for the qubit system, there are twelve MUB fiducial states, which coincide with the H-type magic states. We compare MUB fiducial states and SIC-POVM fiducial states from the perspective of magic resource for stabilizer quantum computation. We further pose the challenging issue of identifying all MUB fiducial states in general dimensions.

Layer codes

Quantum computers require memories that are capable of storing quantum information reliably for long periods of time. The surface code is a two-dimensional quantum memory with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here we present a family of three dimensional topological codes with optimal scaling code parameters and a polynomial energy barrier. Our codes are based on a construction that takes in a stabilizer code and outputs a three-dimensional topological code with related code parameters. The output codes are topological defect networks formed by layers of surface code joined along one-dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good quantum low-density parity-check codes the output codes have optimal scaling. Our results uncover strongly-correlated states of quantum matter that are capable of storing quantum information with the strongest possible protection from errors that is achievable in three dimensions.

High Stability Code Tracking for Band-Limited DSSS Systems

High Stability Code Tracking for Band-Limited DSSS Systems

Qudit stabilizer codes, CFTs, and topological surfaces

We study general maps from the space of rational conformal field theories (CFTs) with a fixed chiral algebra and associated Chern-Simons (CS) theories to the space of qudit stabilizer codes with a fixed generalized Pauli group. We consider certain natural constraints on such a map and show that the map can be described as a graph homomorphism from an orbifold graph, which captures the orbifold structure of CFTs, to a code graph, which captures the structure of self-dual stabilizer codes. By studying explicit examples, we show that this graph homomorphism cannot always be a graph embedding. However, we construct a physically motivated map from universal orbifold subgraphs of CFTs to operators in a generalized Pauli group. We show that this map results in a self-dual stabilizer code if and only if the surface operators in the bulk CS theories corresponding to the CFTs in question are self-dual. For CFTs admitting a stabilizer code description, we show that the full Abelianized generalized Pauli group can be obtained from twisted sectors of certain 0-form symmetries of the CFT. Finally, we connect our construction with SymTFTs, and we argue that many equivalences between codes that arise in our setup correspond to equivalence classes of bulk topological surfaces under fusion with invertible surfaces. Published by the American Physical Society 2024

On the stabilizer formalism and its generalization

Abstract The standard stabilizer formalism provides a setting to show that quantum computations restricted to operations within the Clifford group are classically efficiently simulable: this is the content of the well-known Gottesman–Knill theorem. This work analyzes the mathematical structure behind this theorem to find possible generalizations and derivation of constraints required for constructing a non-trivial generalized Clifford group. We prove that if the closure of the stabilizing set is dense in the set of SU(d) transformations, then the associated Clifford group is trivial, consisting only of local gates and permutations of subsystems. This result demonstrates the close relationship between the density of the stabilizing set and the simplicity of the corresponding Clifford group. We apply the analysis to investigate stabilization with binary observables for qubits and find that the formalism is equivalent to the standard stabilization for a low number of qubits. Based on the observed patterns, we conjecture that a large class of generalized stabilizer states are equivalent to the standard ones. Our results provide better insights into the structure of Gottesman–Knill-type results, consequently allowing us to draw a sharper line between quantum and classical computation.

Fault-tolerant double-circular connectivity pattern for quantum stabilizer codes

Recently, the circular connectivity pattern has been presented for a class of stabilizer quantum error correction codes. The circular connectivity pattern for such a class of stabilizer codes can be implemented in a resource-efficient manner using a single ancilla and native two-qubit Controlled-Not-Swap gates (CNS) gates, which may be interesting for demonstrating error-correction codes with superconducting quantum processors. However, one concern is that this scheme is not fault-tolerant. And it might not apply to the Calderbank-Shor-Steane (CSS) codes. In this paper, we present a fault-tolerant version of the circular connectivity pattern, named the double-circular connectivity pattern. This pattern is an implementation for syndrome-measurement circuits with a flagged error correction scheme for stabilizer codes. We illustrate that this pattern is available for Steane code (a CSS code), Laflamme’s five-qubit code, and Shor’s nine-qubit code. For Laflamme’s five-qubit code and Shor’s nine-qubit code, the pattern has the property that it uses only native two-qubit CNS gates, which are more efficient in the superconducting quantum platform.

Hybrid Stabilizer Matrix Product Operator.

We introduce a novel hybrid approach combining tensor network methods with the stabilizer formalism to address the challenges of simulating many-body quantum systems. By integrating these techniques, we enhance our ability to accurately model unitary dynamics while mitigating the exponential growth of entanglement encountered in classical simulations. We demonstrate the effectiveness of our method through applications to random Clifford T-doped circuits and random Clifford Floquet dynamics. This approach offers promising prospects for advancing our understanding of complex quantum phenomena and accelerating progress in quantum simulation.

Optimality of the Howard-Vala T-gate in stabilizer quantum computation

In a remarkable work [Phys. Rev. A 86 022316 (2012)], Howard and Vala introduced a qudit version of the qubit T-gate (i.e., π/8-gate) for any prime dimensional system. This non-Clifford gate is a key ingredient of the paradigm ‘Clifford +T’, which are widely employed in the stabilizer formalism of universal and fault-tolerant quantum computation. Considering the applications and significance of the T-gate, it is desirable to characterize it from various angles. Here we prove that in any prime dimensional system, the Howard-Vala T-gate is optimal, among all diagonal gates, for generating magic resources from stabilizer states when the magic is quantified via the L 1-norm of characteristic functions (Fourier transforms) of quantum states. The quadratic Gaussian sum in number theory plays a key role in establishing this optimality. This highlights an extreme feature of the Howard-Vala T-gate. We further reveal an intrinsic relation between the Howard-Vala T-gate and the Watson-Campbell-Anwar-Browne T-gate [Phys. Rev. A 92 022312 (2015)] for any prime dimensional system.

Weight-Reduced Stabilizer Codes with Lower Overhead

Stabilizer codes are the most widely studied class of quantum error-correcting codes and form the basis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set of parity-check operators, which are measured in order to infer information about errors that may have occurred. In typical settings, measuring these operators is itself a noisy process and the noise strength scales with the number of qubits involved in a given parity check, or its weight. Hastings has proposed a method for reducing the weights of the parity checks of a stabilizer code, though it has previously only been studied in the asymptotic regime. Here, we instead focus on the regime of small to medium-size codes suitable for quantum computing hardware. We both provide a fully explicit description of Hastings’s method and propose a substantially simplified weight-reduction method that is applicable to the class of quantum product codes. Our simplified method allows us to reduce the check weights of hypergraph- and lifted-product codes to at most six, while preserving the number of logical qubits and at least retaining (in fact, often increasing) the code distance. The price we pay is an increase in the number of physical qubits by a constant factor but we find that our method is much more efficient than Hastings’s method in this regard. We benchmark the performance of our codes on a simulated photonic quantum computing architecture based on Gottesman-Kitaev-Preskill qubits and passive linear optics, finding that our weight-reduction method substantially improves code performance. Published by the American Physical Society 2024

On the Exploration of Quantum Polar Stabilizer Codes and Quantum Stabilizer Codes with High Coding Rate.

Inspired by classical polar codes, whose coding rate can asymptotically achieve the Shannon capacity, researchers are trying to find their analogs in the quantum information field, which are called quantum polar codes. However, no one has designed a quantum polar coding scheme that applies to quantum computing yet. There are two intuitions in previous research. The first is that directly converting classical polar coding circuits to quantum ones will produce the polarization phenomenon of a pure quantum channel, which has been proved in our previous work. The second is that based on this quantum polarization phenomenon, one can design a quantum polar coding scheme that applies to quantum computing. There are several previous work following the second intuition, none of which has been verified by experiments. In this paper, we follow the second intuition and propose a more reasonable quantum polar stabilizer code construction algorithm than any previous ones by using the theory of stabilizer codes. Unfortunately, simulation experiments show that even the stabilizer codes obtained from this more reasonable construction algorithm do not work, which implies that the second intuition leads to a dead end. Based on the analysis of why the second intuition does not work, we provide a possible future direction for designing quantum stabilizer codes with a high coding rate by borrowing the idea of classical polar codes. Following this direction, we find a class of quantum stabilizer codes with a coding rate of 0.5, which can correct two of the Pauli errors.